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Hedonic pricing of cryptocurrency tokens

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Abstract

A cryptocurrency token offers a method of incentivizing behavior in a way that supports trusted interaction (through its blockchain-based infrastructure). It also acts as a multipurpose instrument that may fulfill a variety of roles, such as facilitating digital use cases or acting as a store of value. Understanding how to value such an instrument is complicated by these multiple roles because the relative valuation of one role cannot be disentangled from another role—a token is a ‘bundled’ good. In this work a general pricing model for cryptocurrency tokens is derived, based upon and extending the hedonic pricing framework of Rosen (J Polit Econ 82(1):34–55, https://doi.org/10.1086/260169, 1974) in a partial equilibrium framework. It is shown that individual roles (or characteristics) of a token may be priced by inverting in a special way the relationship between the token’s aggregate quantity and its provision of characteristics. Interaction between a monopolistic token seller and a representative buyer results in an equilibrium that clears both the aggregate token market and the characteristic market. Particular attention is given to the case in which a token possesses a security role, as this has been a focus of existing discussions regarding the regulation of the cryptocurrency market.

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Notes

  1. It is important that this usage of the word utility is not confused with the general benefit to a buyer of a token, as given by the buyer’s utility function, which will also be heavily referenced in this work. Context should prevent confusion between the two meanings.

  2. Recently, public comments by the US Securities and Exchange Commission have indicated that there is no a priori reason to categorize a token as a security by default, in the same sense as a stock or bond is a security, because tokens may provide other uses (e.g., in a utility capacity). See Hinman (2018).

  3. A ‘representative agent’ thus defined is standard in consumer theory. Note that in spite of being a single representative buyer, this representation does not imply market power—the representative buyer is not a monopsonist as it is shorthand for a large number of price-taking, individual buyers.

  4. Where a gender-specific pronoun is used in this paper, a gender-agnostic meaning is conferred.

  5. Here is meant a benefit that will shortly be proxied using a utility function, as is standard in consumption theory.

  6. This excludes the interesting (but significantly more complicated) case where characteristics are ‘in the eye of the beholder’, leading to subjective assessments that match characteristics to goods.

  7. The monopolist assumption is subject to the usual caveat that the seller must know the (representative) buyer’s demand—in the context of token issuance a multi-stage token generation event (TGE), such as an ICO, could facilitate price discovery. Alternatively, a token auction mechanism could be implemented to build upon market research and construct an estimate of demand. I am grateful to an anonymous referee for underscoring this important ‘How would this work in reality?’ question.

  8. The Moore–Penrose inverse is appealing because it generates a set of (in this case unique) numbers \(\hat{n}\) that minimize the Euclidean norm \(\Vert z - \hat{n} a \Vert\), and acts as an ‘inverse’ for a—for this reason it is also known as a ‘least-squares’ pseudo-inverse. Its application is widespread for matrices and not vectors, as used here. See Ben-Israel and Greville (2003) esp. p. 40 for an in-depth discussion, and Barata and Hussein (2011) for a comprehensive overview.

  9. We will assume in what follows that context defines which vector is a ‘row’ vector and which is a ‘column’ vector when forming inner products, and we shall always place row vectors before column vectors.

  10. The constancy of the marginal utility of money is a convenient assumption here and is often made within the quasi-linear utility specification \(U(z,m) := u(z) + m\), so that money acts as a representative of ‘the rest of consumption’ outside of the token.

  11. The reason the seller selects only the direction and not the magnitude of the characteristic mix is discussed below.

  12. See e.g., What is Ether?, an overview of ether at https://www.ethereum.org; retrieved January 2019.

  13. Alternatively, it could simply be assumed that the buyer and seller both recognize and value only non-zero characteristics provided by the token, extending the common knowledge of characteristics assumption stated in Sect. 2.2. This immediately implies that the admissible mixtures are interior.

  14. This is a consequence of using the Moore–Penrose inverse to break the lock-step selection of characteristics offered and token quantity demanded by the buyer that would otherwise occur if, say, the buyer were restricted only to choose the token quantity \(n_{d}\) demanded, taking the relative supply of characteristics per token a and hence total characteristics consumed \(z = n_{d}a\) as given. See Sect. 2.6 for further details.

  15. The index set \(\mathcal {I}\) thus runs over all mixes which are not proportional to each other. We claim momentarily that this set must be at most countably infinite.

  16. We can also appeal to index theory as in general equilibrium models, and impose conditions upon preferences such that (1) the market-clearing characteristic mixes are locally isolated, and (2) the number of such mixes is odd. Although this is interesting in its own right, we shall adopt the genericity argument to claim local isolatedness and leave the index of the set of equilibria to later analysis.

  17. Of course, as the token market continues to become saturated with new offerings, the model presented here may be extended to allow for, e.g., imperfect competition or (as a limiting/benchmark case) perfect competition.

  18. Consumption x is assumed to take place only in period 2. In contrast to the general model in Sect. 2, the unit of account m is no longer simply a proxy for non-token-related consumption. Rather, we assume instead that period 2 consumption x is financed through holdings of the token and holdings of m as an alternative, safe investment described further below.

  19. https://tether.to.

  20. https://www.circle.com/en/usdc.

  21. We are grateful to an anonymous referee for drawing attention to the stablecoin interpretation of this fixed security rate of return assumption.

  22. Recall that an equilibrium characteristic mix a will have \(\Vert a\Vert = 1\), so that the characteristic mix is a unit vector.

  23. One may perhaps wonder why, since the buyer only values second period consumption x, the marginal utility of consumption \(\partial u_{2}/\partial x\) is not factored into the returns of m and \(z_{k}\). This is because both provide the same marginal utility per unit, and so only the wealth-equivalent comparisons matter for demand.

  24. See e.g., Mas-Colell et al. (1995).

  25. As before n is the total number of tokens supplied—which is here the total number \(n_{d} = a^{-1}z^{*}(q,a,r)\) sold to the buyer in period 1—and the function c has standard properties.

  26. If this were not the case, e.g., if characteristics each had their own separate market, then this argument would not hold. One might then need to suggest that the seller has access to financial markets with a higher risk-free rate than the buyer, to allow the seller to gain positive profits in the buyer’s Case I. But the point here is that because the token is a bundled good, such an asymmetry is not required—a negative rate of return on k, ceteris paribus, does not imply negative profits for the seller.

  27. Put still another way, the buyer could have beliefs over the seller that place non-zero measure on the possibility that the seller chooses \(r > (1+r_{f})q \hat{a}_{k}\), and the argument goes through as the measure goes to zero.

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  1. Shorish Research, Meerbeek, Belgium

    Jamsheed Shorish

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  1. Jamsheed Shorish
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Correspondence to Jamsheed Shorish.

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Thanks are given to the participants of the 2nd Berlin Conference ‘Crypto-Currencies in a Digital Economy’, Krzysztof Paruch (Research Institute for Cryptoeconomics, Vienna University of Economics and Business) and the two anonymous referees for very helpful comments and insights.

Appendix

Appendix

1.1 Derivation of relation (3.17)

Consider the buyer’s problem when \(r = (1+r_{f})q\hat{a}_{k}\), i.e., when the buyer is indifferent between using m or \(z_{k}\) to store wealth for second period consumption—the problem can be suggestively written as

$$\begin{aligned} \begin{aligned}&\max _{z, x, m}\, u_{1}(z_{\sim k}) + \beta u_{2}(x) \\&\text {such that} \\&y \ge q\sum _{i \ne k} \hat{a}_{i} z_{i} + (q\hat{a}_{k}z_{k} + m), \\&(1+r_{f})(q \hat{a}_{k}z_{k} + m) \ge x. \end{aligned} \end{aligned}$$

Thus, the buyer’s problem in this case implies consumption (and hence utility) levels that are exactly equivalent to the problem,

$$\begin{aligned} \begin{aligned}&\max _{z, x, m}\, u_{1}(z_{\sim k}) + \beta u_{2}(x), \\&\text {such that} \\&y \ge q\sum _{i \ne k} \hat{a}_{i} z_{i} + \frac{1}{1+r_{f}} x. \\ \end{aligned} \end{aligned}$$

In particular, second period consumption \(x^{*} = x^{*}(q)\) (suppressing dependence upon \(\hat{a}\)) is fixed once q is given, and the locus of points \((z_{k},m)\) satisfy \(x^{*}(q) = (1+r_{f})q\hat{a}_{k}z_{k} + m\), with \(z_{k} \in [0,x^{*}/(q\hat{a}_{k})]\) and \(m \in [0, x^{*}]\).

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Shorish, J. Hedonic pricing of cryptocurrency tokens. Digit Finance 1, 163–189 (2019). https://doi.org/10.1007/s42521-019-00005-y

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